StratonovichProcess

StratonovichProcess[{a,b},x,t]

represents a Stratonovich process , where .

StratonovichProcess[{a,b,c},x,t]

represents a Stratonovich process , where .

StratonovichProcess[,,{x,x0},{t,t0}]

represents a Stratonovich process with initial condition .

StratonovichProcess[,,,Σ]

uses a Wiener process , with covariance Σ.

StratonovichProcess[proc]

converts proc to a standard Stratonovich process whenever possible.

StratonovichProcess[sdeqns,expr,x,t,wdproc]

represents a Stratonovich process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc.

Details and Options

  • StratonovichProcess is also known as Stratonovich diffusion or stochastic differential equation (SDE).
  • StratonovichProcess is a continuous-time and continuous-state random process.
  • If the drift a is an -dimensional vector and the diffusion b an ×-dimensional matrix, the process is -dimensional and driven by an -dimensional WienerProcess.
  • Common specifications for coefficients a and b include:
  • a scalar, b scalar
    a scalar, b vector
    a vector, b vector
    a vector, b matrix
  • A stochastic differential equation is sometimes written as an integral equation .
  • The default initial time t0 is taken to be zero, and default initial state x0 is zero.
  • The default covariance Σ is the identity matrix.
  • A standard Stratonovich process has output , consisting of a subset of differential states .
  • Processes proc that can be converted to standard StratonovichProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, ItoProcess, and StratonovichProcess.
  • The stochastic differential equations in sdeqns can be of the form , where is \[DifferentialD], which can be input using dd. The differentials and are taken to be Stratonovich differentials.
  • The output expression expr can be any expression involving x[t] and t.
  • The driving process dproc can be any process that can be converted to a standard Stratonovich process.
  • Method settings in RandomFunction specific to StratonovichProcess include:
  • "EulerMaruyama"EulerMaruyama (order 1/2, default)
    "KloedenPlatenSchurz"KloedenPlatenSchurz (order 3/2)
    "Milstein"Milstein (order 1)
    "StochasticRungeKutta"3stage Rossler SRK scheme (order 1)
    "StochasticRungeKuttaScalarNoise"3stage Rossler SRK scheme for scalar noise (order 3/2)
  • StratonovichProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.

Examples

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Basic Examples  (1)

Define a process by its stochastic differential equation:

Simulate the process:

Compute the mean function:

Compute the covariance function:

Scope  (14)

Basic Uses  (9)

Define a Wiener process with drift and diffusion from the sde :

Directly convert from the parametric process:

Define a process , where :

Use differential notation to define the same process:

Define a vector process with output :

Using differential notation:

Define a vector process , where :

Using differential notation:

Define a vector process where :

Using differential notation:

Define a process driven by two correlated Wiener processes:

Define a scalar process corresponding to the sde :

Define vector process and corresponding to the sde and :

Define a process corresponding to the 2D correlated Wiener process:

Define a vector process driven by the correlated 2D Wiener process:

Define a Stratonovich process by its stochastic differential equation:

Drift and diffusion of the process:

Kolmogorov forward equation:

Inactive is used here to avoid expanding the partial derivatives. Use Activate to expand the expression:

Kolmogorov backward equation:

Compute the Stratonovich derivative of a function . The output is a list consisting of drift and diffusion terms:

Special Stratonovich Processes  (5)

A Stratonovich process corresponding to the WienerProcess:

A Stratonovich process corresponding to the GeometricBrownianMotionProcess:

A Stratonovich process corresponding to the BrownianBridgeProcess:

A Stratonovich process corresponding to the OrnsteinUhlenbeckProcess:

A Stratonovich process corresponding to the CoxIngersollRossProcess:

Applications  (1)

Define a vector process corresponding to iterated Stratonovich integrals , , , , , :

Compute its mean function:

And its covariance function:

Properties & Relations  (1)

Convert ItoProcess to StratonovichProcess:

Convert back:

Possible Issues  (2)

StratonovichProcess does not support random initial conditions, so cannot be represented:

But it supports processes with fixed initial condition:

Initial time of the driven process needs to match with StratonovichProcess:

With matching initial time, this can be represented:

Introduced in 2012
 (9.0)